Problem: Suppose we have a vector-valued function $g(t)$ and a scalar function $f(x, y, z)$. Let $h(t) = f(g(t))$. We know: $\begin{aligned} &g(4) = (-2, 3, 1) \\ \\ &g'(4) = (8, 1, 3) \\ \\ &\nabla f(-2, 3, 1) = (0, 6, 2) \end{aligned}$ Evaluate $\dfrac{d h}{d t}$ at $t = 4$. $h'(4)=$
Explanation: Formula The multivariable chain rule says that $\dfrac{dh}{dt} = \nabla f(g(t)) \cdot g'(t)$. The $g'(t)$ part is how much a change in $t$ will cause the input to $f$ to move, and the $\nabla f(g(t))$ part is how much $f$ will change in response to this update to its input. [What's the intuition behind the formula?] Applying the formula We want to find $h'(4) = \nabla f(g(4)) \cdot g'(4)$. We know the following. $\begin{aligned} &g(4) = (-2, 3, 1) \\ \\ &g'(4) = (8, 1, 3) \\ \\ &\nabla f(-2, 3, 1) = (0, 6, 2) \end{aligned}$ Substituting: $h'(4) = (0, 6, 2) \cdot (8, 1, 3) = 12$ Answer Therefore, $h'(4) = 12$.